Abstract. In a series of works [18,21,19,20,23,22], Geiß-Leclerc-Schröer defined the cluster algebra structure on the coordinate ring C[N (w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig's dual semicanonical basis S * . We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras in [4] to the dual canonical basis B up . In particular, we prove that the quantum analogue Oq[N (w)] of C[N (w)] has the induced basis from B up , which contains quantum flag minors and satisfies a factorization property with respect to the 'q-center' of Oq[N (w)]. This generalizes Caldero's results [7,8,9] from ADE cases to an arbitary symmetrizable Kac-Moody Lie algebra.

Abstract. Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra.In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez-Leclerc from the Kronecker and Dynkin quiver case to the acyclic case.The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization.

In this paper, we construct twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells.Moreover we prove that quantum twist automorphisms are described by the syzygy functors for representations of preprojective algebras in the symmetric case. This is the quantum analogue of Geiß-Leclerc-Schröer's description, and Geiß-Leclerc-Schröer's results are essential in our proof. As a consequence, we show that quantum twist automorphisms are compatible with quantum cluster monomials. The 6-periodicity of specific quantum twist automorphisms is also verified.

Abstract. In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincaré-Birkhoff-Witt type bases under the "reverse" lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors.

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